Mathematical skills in Psychology | AQA A-Level Psychology Revision
- Revision Notes
- Aug 5
- 17 min read
Updated: 5 days ago
For 7182 specification, first teach in September 2025
AQA A-Level Psychology | Free Revision Notes
Estimated study time: 60 minutes
This Mathematical skills in Psychology A-Level revision page explains how numerical techniques are applied to psychological research. You will learn to estimate results, make order-of-magnitude calculations, substitute data into statistical equations and solve simple equations. Mathematical questions are always placed within a psychological context, so you need to understand both the calculation and what the result represents. This lesson brings together skills from percentages, ratios and numerical data, descriptive statistics and inferential testing.
Learning Objectives 🎯
By the end of this revision page, you should be able to:
Estimate psychological research results.
Make order-of-magnitude calculations.
Use rounding to check whether an answer is reasonable.
Substitute numerical values into statistical equations.
Use appropriate mathematical symbols and units.
Solve simple equations used in psychological research.
Rearrange simple equations to find an unknown value.
Interpret a calculated result in the context of a study.
Revision Notes 📚
Mathematical skills in A-Level Psychology
Mathematical skills are an important part of AQA A-Level Psychology.
At least \(10\%\) of the marks in the assessment require mathematical skills at Higher Tier GCSE Mathematics standard or above.
These skills are applied to psychological research rather than tested as isolated mathematics.
You may be asked to:
estimate a result;
calculate a percentage;
calculate or interpret a mean;
use ratios and fractions;
convert between decimal and standard form;
use significant figures;
substitute values into an equation;
solve a simple equation;
calculate degrees of freedom;
interpret probability;
select and use a statistical test;
translate information between numerical, graphical and algebraic forms.
A strong answer should show both accurate working and an understanding of what the result means.
The three main skills in this lesson
This lesson focuses on three connected areas.
Estimating results
Estimation involves using rounded or approximate values to produce a reasonable answer.
Substituting values
Substitution involves replacing the letters or symbols in an equation with the numerical values from a research study.
Solving equations
Solving an equation involves finding an unknown value while keeping both sides of the equation balanced.
These skills may be combined in one examination question.
Estimating results
What is estimation?
An estimate is an approximate value rather than an exact result.
Estimation may be used to:
make a quick calculation;
identify the approximate size of a result;
check whether an exact answer is reasonable;
interpret data presented in a graph;
compare the sizes of two results;
calculate from rounded values.
The AQA specification gives estimating the range of a set of scores as one possible psychological example.
Exact and estimated answers
Suppose the exact values are:
$$47.8+31.6$$
The exact answer is:
$$79.4$$
For an estimate, the values could be rounded:
$$47.8\approx50$$
$$31.6\approx30$$
Therefore:
$$50+30=80$$
The estimated answer is:
$$\boxed{80}$$
This is close to the exact result of \(79.4\), so the exact calculation appears reasonable.
Why psychologists estimate results
A researcher may need to estimate:
the approximate mean for a large sample;
the approximate range shown by a graph;
the likely percentage within a category;
whether a statistical result is of the expected size;
whether a calculator answer contains a keying error.
Estimation is particularly useful when a question asks whether a result is plausible.
Rounding before estimating
A common estimation method is to round each value to one significant figure.
For example:
$$398\times21$$
Round the values:
$$398\approx400$$
$$21\approx20$$
Estimate:
$$400\times20=8000$$
The exact answer should therefore be reasonably close to:
$$8000$$
Estimating a division
Suppose a researcher obtains a total score of:
$$5968$$
from:
$$203$$
participants.
To estimate the mean, round the values:
$$5968\approx6000$$
$$203\approx200$$
Then calculate:
$$\frac{6000}{200}=30$$
The estimated mean is:
$$\boxed{30}$$
An exact answer that was extremely different from \(30\) would need to be checked.
Estimating a mean from a large total
The mean is calculated using:
$$\text{Mean}=\frac{\sum x}{N}$$
Suppose the total score for a large group is:
$$48\,360$$
and the sample contains:
$$2010$$
participants.
Round the values:
$$48\,360\approx50\,000$$
$$2010\approx2000$$
Substitute the estimated values:
$$\text{Estimated mean}=\frac{50\,000}{2000}$$
$$\text{Estimated mean}=25$$
The exact mean should be approximately:
$$25$$
This is an example of making an order-of-magnitude calculation from a large overall score.
The exact calculation of the mean is covered in measures of central tendency.
Estimating a range
The range is calculated using:
$$\text{Range}=\text{Highest score}-\text{Lowest score}$$
Suppose a graph suggests that:
the highest score is approximately \(48\);
the lowest score is approximately \(12\).
The estimated range is:
$$48-12=36$$
If the values are difficult to read precisely, they could be rounded:
$$50-10=40$$
The estimated range is then:
$$\boxed{40}$$
The range and standard deviation are covered in measures of dispersion.
Estimating percentages
Suppose approximately \(49\) out of \(101\) participants selected an option.
The percentage is:
$$\frac{49}{101}\times100$$
For an estimate:
$$49\approx50$$
$$101\approx100$$
Therefore:
$$\frac{50}{100}\times100=50\%$$
The estimated percentage is:
$$\boxed{50\%}$$
Checking a calculated percentage
Suppose \(18\) out of \(60\) participants selected a response.
The exact percentage is:
$$\frac{18}{60}\times100=30\%$$
A quick estimate can be made by recognising that:
$$18\approx20$$
and:
$$\frac{20}{60}=\frac{1}{3}$$
One third is approximately:
$$33\%$$
The exact answer of \(30\%\) is close to the estimate and therefore appears reasonable.
Estimating from a graph
When estimating from a graph:
Read the axis carefully.
Identify the scale interval.
Locate the position of the value between marked points.
State that the result is approximate.
Include units where appropriate.
For example, if a bar is positioned approximately halfway between:
$$20$$
and:
$$30$$
a reasonable estimate is:
$$25$$
Do not report more precision than the graph allows.
Order-of-magnitude calculations
What is an order of magnitude?
An order of magnitude describes the approximate scale or size of a value.
It may be expressed using powers of ten.
For example:
$$10^1=10$$
$$10^2=100$$
$$10^3=1000$$
$$10^4=10\,000$$
Order-of-magnitude thinking helps a researcher identify whether a result is:
in the tens;
in the hundreds;
in the thousands;
much smaller than \(1\).
Psychological example
Suppose a research database contains results from:
$$784$$
participants.
This is of the approximate order:
$$10^3$$
because the number is on the scale of hundreds approaching one thousand.
If a database contains:
$$12\,400$$
scores, it is on the approximate scale of:
$$10^4$$
Order of magnitude and standard form
Standard form is written as:
$$a\times10^n$$
where:
$$1\leq a<10$$
For example:
$$48\,000=4.8\times10^4$$
and:
$$0.0062=6.2\times10^{-3}$$
The power of ten shows the scale of the value.
Recognising decimal and standard form is developed in percentages, ratios and numerical data.
Estimating using one significant figure
Suppose a psychologist records a total of:
$$97\,840$$
responses from:
$$3980$$
participants.
Round each value to one significant figure:
$$97\,840\approx100\,000$$
$$3980\approx4000$$
Calculate the estimated mean:
$$\frac{100\,000}{4000}=25$$
The estimated mean is:
$$\boxed{25}$$
This calculation provides the approximate scale of the mean before an exact calculation is attempted.
Checking an order-of-magnitude error
Suppose a student calculates:
$$\frac{48\,000}{2000}=240$$
An estimate shows that this cannot be correct:
$$\frac{50\,000}{2000}=25$$
The correct answer should be in the tens, not the hundreds.
The likely error is an extra zero.
Order-of-magnitude checking is valuable because calculator errors may otherwise look convincing.
Estimating multiplication
Suppose a researcher expects:
$$198$$
participants to complete:
$$31$$
trials each.
Estimate the number of trial results:
$$198\approx200$$
$$31\approx30$$
Therefore:
$$200\times30=6000$$
The researcher should expect approximately:
$$6000$$
trial results.
Estimating the number in a category
Suppose a researcher expects approximately:
$$19\%$$
of:
$$510$$
participants to select a response.
Round the values:
$$19\%\approx20\%$$
$$510\approx500$$
Then:
$$20\%\text{ of }500=\frac{20}{100}\times500$$
$$=100$$
Approximately \(100\) participants are expected in the category.
Reasonableness checks
After completing a calculation, ask:
Is the answer approximately the size expected?
Is the percentage between \(0\%\) and \(100\%\)?
Is the mean between the lowest and highest scores?
Is a frequency a whole number?
Is the number of participants greater than zero?
Are the units appropriate?
Is the answer consistent with the graph or table?
These checks can identify errors without repeating the entire calculation.
Substituting values into equations
What is substitution?
Substitution means replacing the letters or symbols in an equation with the numerical values provided.
For example, the equation:
$$\text{Mean}=\frac{\sum x}{N}$$
contains:
\(\sum x\), the total of the scores;
\(N\), the number of scores.
If:
$$\sum x=240$$
and:
$$N=12$$
substitute these values:
$$\text{Mean}=\frac{240}{12}$$
Then calculate:
$$\boxed{\text{Mean}=20}$$
A reliable substitution method
Use the following steps.
Write the equation.
Identify what each symbol represents.
Select the correct values from the question.
Include the values in the correct positions.
Use brackets where necessary.
Calculate in the correct order.
Include appropriate units.
Round only when needed.
Interpret the answer in context.
Why writing the equation helps
Writing the equation before inserting values:
makes the method clear;
reduces the risk of using the wrong values;
allows working marks to be awarded;
helps identify units;
makes the answer easier to check.
A calculator answer without working may not show the method used.
Substitution into the mean equation
Mean equation
The mean is calculated using:
$$\bar{x}=\frac{\sum x}{N}$$
where:
\(\bar{x}\) is the mean;
\(\sum x\) is the sum of the scores;
\(N\) is the number of scores.
Worked example
A memory study produces a total score of:
$$286$$
from:
$$13$$
participants.
Substitute the values:
$$\bar{x}=\frac{286}{13}$$
Calculate:
$$\bar{x}=22$$
Therefore:
$$\boxed{\bar{x}=22\text{ words}}$$
The unit is included because the score represents words recalled.
Checking the mean
If individual scores range from:
$$15$$
to:
$$28$$
a mean of:
$$22$$
is plausible because it lies between the lowest and highest scores.
A result such as:
$$220$$
would be unreasonable and should be checked.
Substitution into a percentage equation
Percentage equation
A percentage can be calculated using:
$$\text{Percentage}=\frac{\text{Frequency}}{\text{Total}}\times100$$
Worked example
In an observation study:
\(18\) instances were classified as cooperative;
\(60\) behaviours were recorded in total.
Substitute the values:
$$\text{Percentage cooperative}=\frac{18}{60}\times100$$
Calculate:
$$\text{Percentage cooperative}=30\%$$
Therefore:
$$\boxed{30\%}$$
Checking the answer
The result must lie between:
$$0\%$$
and:
$$100\%$$
A result of:
$$300\%$$
would indicate an error in the substitution or calculation.
Substitution into an expected-frequency equation
Expected-frequency equation
For a Chi-squared test:
$$E=\frac{\text{Row total}\times\text{Column total}}{\text{Overall total}}$$
where:
\(E\) is the expected frequency;
the row total comes from the relevant row;
the column total comes from the relevant column;
the overall total is the number of cases in the complete table.
Worked example
Suppose:
$$\text{Row total}=40$$
$$\text{Column total}=30$$
$$\text{Overall total}=60$$
Substitute:
$$E=\frac{40\times30}{60}$$
Calculate the numerator:
$$40\times30=1200$$
Then divide:
$$E=\frac{1200}{60}$$
$$\boxed{E=20}$$
The calculation of expected frequencies is covered in Chi-squared test.
Using brackets correctly
Suppose the equation contains:
$$(O-E)^2$$
and:
$$O=18$$
$$E=14$$
Substitute both values inside the brackets:
$$(18-14)^2$$
Calculate the brackets first:
$$4^2$$
Then square:
$$16$$
Writing:
$$18-14^2$$
would be incorrect because the square would be applied only to \(14\).
Substitution into the Chi-squared equation
Chi-squared equation
The equation is:
$$\chi^2=\sum\frac{(O-E)^2}{E}$$
Each table cell contributes a value to the final total.
Worked cell contribution
Suppose:
$$O=18$$
and:
$$E=14$$
Substitute:
$$\frac{(O-E)^2}{E}=\frac{(18-14)^2}{14}$$
Calculate the difference:
$$18-14=4$$
Square the result:
$$4^2=16$$
Divide by the expected frequency:
$$\frac{16}{14}=1.142857\ldots$$
To three significant figures:
$$\boxed{1.14}$$
This is the contribution of one cell, not the complete Chi-squared value.
The contributions from all cells would need to be added.
Order of operations
When substituting into an equation, use the correct order of operations:
Brackets.
Powers.
Division and multiplication.
Addition and subtraction.
For:
$$\frac{(18-14)^2}{14}$$
the order is:
$$18-14=4$$
$$4^2=16$$
$$16\div14=1.14$$
Substitution into degrees of freedom
Degrees-of-freedom equation
For a Chi-squared contingency table:
$$df=(r-1)(c-1)$$
where:
\(r\) is the number of data rows;
\(c\) is the number of data columns.
Worked example
A contingency table has:
$$r=3$$
data rows and:
$$c=4$$
data columns.
Substitute:
$$df=(3-1)(4-1)$$
Calculate the brackets:
$$df=2\times3$$
Therefore:
$$\boxed{df=6}$$
Do not include totals rows or totals columns when identifying \(r\) and \(c\).
Substitution and standard deviation
Standard-deviation equations
A standard-deviation calculation uses:
each score;
the mean;
the number of scores;
deviations from the mean.
The exact equation provided in a question should be followed carefully.
When substituting into a standard-deviation equation:
Calculate the mean if required.
Find each deviation from the mean.
Square each deviation.
Add the squared deviations.
Divide using the denominator shown.
Find the square root.
Standard deviation is covered in measures of dispersion.
Square roots
A square root reverses a square.
For example:
$$\sqrt{81}=9$$
because:
$$9^2=81$$
If an equation requires a square root, make sure it is applied to the complete value inside the root.
For example:
$$\sqrt{\frac{64}{4}}$$
Calculate inside the root first:
$$\sqrt{16}$$
Then:
$$4$$
Solving simple equations
What does solving an equation mean?
Solving an equation means finding the value of an unknown quantity.
An equation must remain balanced.
If the same operation is performed on one side, it must also be performed on the other side.
Psychological research equations may ask you to find:
a missing total;
a missing frequency;
a sample size;
a mean;
a percentage;
a number of participants;
degrees of freedom.
Inverse operations
To solve an equation, use the inverse operation.
Operation | Inverse operation |
Addition | Subtraction |
Subtraction | Addition |
Multiplication | Division |
Division | Multiplication |
Squaring | Square root |
For example, if:
$$x+7=19$$
subtract \(7\) from both sides:
$$x=19-7$$
$$\boxed{x=12}$$
Solving the mean equation
Finding the mean
If:
$$\bar{x}=\frac{\sum x}{N}$$
and the total and sample size are known, divide:
$$\bar{x}=\frac{240}{12}=20$$
Finding the total score
Start with:
$$\bar{x}=\frac{\sum x}{N}$$
Multiply both sides by \(N\):
$$\sum x=\bar{x}\times N$$
Suppose:
$$\bar{x}=16$$
and:
$$N=25$$
Substitute:
$$\sum x=16\times25$$
Calculate:
$$\boxed{\sum x=400}$$
The total of all scores is \(400\).
Finding the number of scores
Start with:
$$\bar{x}=\frac{\sum x}{N}$$
Rearrange to find \(N\):
$$N=\frac{\sum x}{\bar{x}}$$
Suppose:
$$\sum x=540$$
and:
$$\bar{x}=18$$
Substitute:
$$N=\frac{540}{18}$$
Calculate:
$$\boxed{N=30}$$
There were \(30\) scores.
Checking the sample size
A sample size should normally be:
positive;
a whole number;
consistent with the study.
A calculation producing:
$$N=30.6$$
would usually require checking because a study cannot contain a fraction of a participant.
Solving percentage equations
Finding a percentage
Use:
$$\text{Percentage}=\frac{\text{Frequency}}{\text{Total}}\times100$$
Finding the frequency from a percentage
Rearrange:
$$\text{Frequency}=\frac{\text{Percentage}}{100}\times\text{Total}$$
Suppose:
\(35\%\) of participants selected Option A;
there were \(80\) participants.
Substitute:
$$\text{Frequency}=\frac{35}{100}\times80$$
$$\text{Frequency}=0.35\times80$$
$$\boxed{\text{Frequency}=28}$$
Therefore, \(28\) participants selected Option A.
Finding the total from a frequency and percentage
Start with:
$$\text{Percentage}=\frac{\text{Frequency}}{\text{Total}}\times100$$
Rearrange:
$$\text{Total}=\frac{\text{Frequency}\times100}{\text{Percentage}}$$
Suppose:
\(24\) participants represent \(30\%\) of the sample.
Substitute:
$$\text{Total}=\frac{24\times100}{30}$$
$$\text{Total}=\frac{2400}{30}$$
$$\boxed{\text{Total}=80}$$
The sample contained \(80\) participants.
Solving a ratio problem
Suppose participants are divided into two groups in the ratio:
$$3:2$$
The sample contains:
$$50$$
participants.
Add the ratio parts:
$$3+2=5$$
Find the number represented by one part:
$$50\div5=10$$
Calculate each group:
$$3\times10=30$$
$$2\times10=20$$
The groups contain:
$$\boxed{30\text{ and }20\text{ participants}}$$
This skill may be used when interpreting samples or group allocation.
Solving degrees-of-freedom equations
Finding degrees of freedom
Use:
$$df=(r-1)(c-1)$$
If:
$$r=4$$
and:
$$c=3$$
then:
$$df=(4-1)(3-1)$$
$$df=3\times2$$
$$\boxed{df=6}$$
Finding a missing number of columns
Suppose:
$$df=6$$
and:
$$r=3$$
Start with:
$$6=(3-1)(c-1)$$
Simplify:
$$6=2(c-1)$$
Divide both sides by \(2\):
$$3=c-1$$
Add \(1\):
$$\boxed{c=4}$$
The contingency table has four data columns.
Finding a missing number of rows
Suppose:
$$df=8$$
and:
$$c=3$$
Substitute:
$$8=(r-1)(3-1)$$
Simplify:
$$8=2(r-1)$$
Divide by \(2\):
$$4=r-1$$
Add \(1\):
$$\boxed{r=5}$$
The table has five data rows.
Solving probability statements
Converting probability to percentage
To convert a probability into a percentage:
$$\text{Percentage}=p\times100$$
For:
$$p=0.05$$
the percentage is:
$$0.05\times100=5\%$$
For:
$$p=0.01$$
the percentage is:
$$0.01\times100=1\%$$
Converting percentage to probability
Divide the percentage by \(100\):
$$p=\frac{\text{Percentage}}{100}$$
For:
$$5\%$$
$$p=\frac{5}{100}=0.05$$
The interpretation of these values is covered in probability and significance.
Using mathematical symbols
Equality and inequality symbols
You should recognise and use symbols including:
Symbol | Meaning |
\(=\) | Equal to |
\(<\) | Less than |
\(>\) | Greater than |
\(\leq\) | Less than or equal to |
\(\geq\) | Greater than or equal to |
\(\propto\) | Proportional to |
\(\sim\) | Similar to or approximately distributed as, depending on context |
\(\ll\) | Much less than |
\(\gg\) | Much greater than |
Statistical conclusion symbols
A significant result at the \(0.05\) level may be written as:
$$p\leq0.05$$
A significant result at the \(0.01\) level may be written as:
$$p\leq0.01$$
A result that is not statistically significant at the \(0.05\) level may be written as:
$$p>0.05$$
Comparison rules
For some statistical tests, significance requires:
$$\text{Observed value}\geq\text{Critical value}$$
For others, significance requires:
$$\text{Observed value}\leq\text{Critical value}$$
The comparison rule must match the statistical test.
Do not choose the direction of the inequality merely by looking at which value is larger.
Units in psychological calculations
Why units matter
A numerical answer is incomplete when the quantity has a unit.
Possible units include:
seconds;
milliseconds;
centimetres;
words recalled;
errors;
participants;
percentage;
scale points.
For example:
$$\bar{x}=18$$
is less informative than:
$$\bar{x}=18\text{ words recalled}$$
Units within equations
When substituting physical quantities into an equation, use compatible units.
For example, do not combine:
$$2\text{ minutes}$$
with:
$$30\text{ seconds}$$
without converting them into the same unit.
Convert:
$$2\text{ minutes}=120\text{ seconds}$$
Then the values can be compared or combined.
Converting time units
Use:
$$1\text{ minute}=60\text{ seconds}$$
For example:
$$3.5\text{ minutes}=3.5\times60$$
$$\boxed{210\text{ seconds}}$$
Converting milliseconds
Use:
$$1000\text{ milliseconds}=1\text{ second}$$
For example:
$$2500\text{ milliseconds}=\frac{2500}{1000}$$
$$\boxed{2.5\text{ seconds}}$$
Significant figures and decimal places
Significant figures
The first significant figure is the first non-zero digit.
For example:
$$0.006284$$
To two significant figures:
$$0.0063$$
To three significant figures:
$$0.00628$$
Decimal places
Decimal places count the digits after the decimal point.
For:
$$4.2867$$
to two decimal places:
$$4.29$$
Avoid premature rounding
Keep the full calculator value during intermediate stages.
Round only the final answer unless the question instructs otherwise.
Suppose a calculation produces:
$$1.142857\ldots$$
If this value will be used in a later calculation, retain the full calculator value.
Round the final answer to the required number of significant figures.
Reporting correlation coefficients
Correlation coefficients should be reported using an appropriate number of significant figures, such as two or three significant figures.
For example:
$$r_s=0.73642\ldots$$
to three significant figures becomes:
$$\boxed{r_s=0.736}$$
Translating a mathematical result into psychology
Calculating is only part of the answer
A mathematical result should often be interpreted in the context of the study.
For example, after calculating:
$$\bar{x}=14.2$$
a complete statement might be:
Participants recalled a mean of \(14.2\) words.
After calculating:
$$35\%$$
a complete statement might be:
\(35\%\) of participants selected the behavioural response.
Interpreting an estimate
If the estimated mean is:
$$25$$
you might write:
The average test score is expected to be approximately \(25\) marks.
The word approximately makes it clear that the result was estimated.
Interpreting degrees of freedom
If:
$$df=2$$
a suitable statement is:
The researcher should use the row for \(df=2\) in the Chi-squared critical-values table.
Interpreting an expected frequency
If:
$$E=20$$
a suitable statement is:
If there were no association between the variables, \(20\) cases would be expected in this category combination.
Translating between forms
AQA may require you to translate information between:
numerical form;
graphical form;
algebraic form.
For example, you might:
use raw scores to construct a bar chart;
read a frequency from a graph and insert it into an equation;
calculate a percentage and display it in a table;
use a contingency table to calculate expected frequencies;
interpret an equation in words.
These skills are developed further in translating and presenting data.
A complete approach to mathematical questions
Step 1: Identify what the question asks
Look for command words such as:
calculate;
estimate;
state;
show;
determine;
use;
explain;
interpret.
An estimate does not require the same precision as an exact calculation.
Step 2: Identify the information provided
Underline or list:
numerical values;
units;
totals;
sample sizes;
row and column totals;
required significance levels.
Step 3: Select the equation
Choose the equation that matches the required quantity.
Do not select an equation simply because it contains familiar symbols.
Step 4: Write the equation
For example:
$$\text{Percentage}=\frac{\text{Frequency}}{\text{Total}}\times100$$
Step 5: Substitute carefully
Replace each symbol with the relevant value:
$$\text{Percentage}=\frac{18}{60}\times100$$
Step 6: Calculate in the correct order
Use brackets and powers before division, multiplication, addition and subtraction.
Step 7: Check the order of magnitude
Estimate what the answer should be and compare it with the exact result.
Step 8: Include units and precision
Use:
appropriate units;
significant figures;
decimal places;
percentage symbols.
Step 9: Interpret the result
Explain what the number represents within the psychological investigation.
Key Words 🔑
Key word | Student-friendly definition | How it may be used in an exam |
Estimate | An approximate value calculated using rounded information. | You may estimate a range, mean, percentage or total. |
Order of magnitude | The approximate scale or size of a number, often described using powers of ten. | You may use it to check whether a calculated answer is reasonable. |
Rounding | Replacing a value with a nearby value containing fewer digits. | Values may be rounded before making an estimate. |
Significant figure | A digit contributing to the precision of a number, beginning with the first non-zero digit. | Statistical values may need to be reported to two or three significant figures. |
Standard form | A way of writing a number as \(a\times10^n\), where \(1\leq a<10\). | It may be used to express very large or small psychological data values. |
Substitution | Replacing symbols in an equation with numerical values. | You may substitute data into a statistical equation. |
Equation | A mathematical statement showing that two expressions are equal. | You may solve or rearrange an equation to find an unknown quantity. |
Unknown | A value represented by a letter or symbol that must be found. | You may solve for a missing total, frequency or sample size. |
Rearrange | Change the form of an equation while preserving its equality. | You may rearrange the mean or percentage equation. |
Inverse operation | An operation that reverses another operation. | Division reverses multiplication, while a square root reverses squaring. |
Mean | The total of the scores divided by the number of scores. | You may estimate it or solve the equation for a missing value. |
Expected frequency | The frequency predicted if there is no association between categorical variables. | You may substitute row, column and overall totals into its equation. |
Degrees of freedom | A value used to select a row from some statistical tables. | For Chi-squared, calculate it using \((r-1)(c-1)\). |
Observed value | A statistical value calculated from the research data. | It may be compared with a critical value. |
Critical value | A threshold obtained from a statistical table. | It is used to determine statistical significance. |
Unit | The measurement attached to a numerical value. | Answers may require seconds, milliseconds, participants or percentages. |
Reasonableness check | A check that a result is consistent with the approximate size and context of the data. | Estimation can reveal calculator or substitution errors. |
Hints from the Examiner Reports 💡
No lesson-specific examiner guidance was identified in the provided reports.
Common Mistakes ⚠️
Mistake: Giving an exact answer when the question asks for an estimate.
Why this is incorrect:
An estimation question is testing whether you can round values and identify the approximate size of a result.
How to improve:
Round the values sensibly before calculating and use words such as approximately.
Mistake: Rounding values inconsistently.
Why this is incorrect:
Rounding one number substantially while leaving another highly precise can make an estimate harder to calculate and less useful.
How to improve:
Round the main values to a similar level of precision, commonly one significant figure.
Mistake: Accepting a calculator answer without checking its size.
Why this is incorrect:
A mistyped value or misplaced decimal point can produce a mathematically valid-looking but unreasonable answer.
How to improve:
Make an order-of-magnitude estimate before or after the exact calculation.
Mistake: Reporting a mean outside the range of the data.
Why this is incorrect:
The arithmetic mean should lie between the lowest and highest scores.
How to improve:
Compare the calculated mean with the original scores.
Mistake: Substituting a value into the wrong part of an equation.
Why this is incorrect:
Each symbol has a specific meaning.
For example, in:
$$E=\frac{\text{Row total}\times\text{Column total}}{\text{Overall total}}$$
the overall total must be placed in the denominator.
How to improve:
Define each symbol before substituting.
Mistake: Omitting brackets.
Why this is incorrect:
Without brackets, operations may be completed in the wrong order.
For example:
$$(18-14)^2$$
is not the same as:
$$18-14^2$$
How to improve:
Copy the structure of the original equation before inserting values.
Mistake: Squaring only one part of a difference.
Why this is incorrect:
In:
$$(O-E)^2$$
the complete difference between observed and expected frequencies must be squared.
How to improve:
Calculate the value inside the brackets first.
Mistake: Rearranging an equation by changing only one side.
Why this is incorrect:
An equation must remain balanced.
How to improve:
Perform the same operation on both sides.
Mistake: Multiplying when division is required.
Why this is incorrect:
For the mean:
$$\bar{x}=\frac{\sum x}{N}$$
the total is divided by the number of scores.
How to improve:
Write the equation before calculating.
Mistake: Giving a decimal number of participants.
Why this is incorrect:
A participant frequency must normally be a whole number.
How to improve:
Check the calculation and whether rounding is permitted.
Mistake: Omitting units.
Why this is incorrect:
The result may be ambiguous without seconds, milliseconds, words, participants or another appropriate unit.
How to improve:
Return to the variable named in the question and attach its unit.
Mistake: Mixing units within a calculation.
Why this is incorrect:
Values measured in minutes and seconds cannot be combined directly.
How to improve:
Convert all values to the same unit before substituting.
Mistake: Rounding during every intermediate stage.
Why this is incorrect:
Repeated rounding can make the final answer less accurate.
How to improve:
Retain full calculator values and round the final answer only.
Mistake: Confusing decimal places with significant figures.
Why this is incorrect:
Decimal places count digits after the decimal point. Significant figures begin with the first non-zero digit.
How to improve:
Identify which form of rounding the question requests.
Mistake: Completing a calculation without interpreting it.
Why this is incorrect:
A psychological research question may ask what the value means, not only what it equals.
How to improve:
Write a sentence linking the result to the participants, behaviour or variable.
Exam-Style Questions ✍️
Question 1
A researcher records a total score of:
$$3980$$
from:
$$102$$
participants.
Estimate the mean score by rounding each value to one significant figure. Show your working.[2 marks]
Question 2
A psychologist records response times ranging from approximately:
$$12\text{ seconds}$$
to:
$$48\text{ seconds}$$
Estimate the range.[2 marks]
Question 3
A study contains:
$$198$$
participants, each completing:
$$21$$
trials.
Estimate the total number of trial results. Show how you rounded the values.[2 marks]
Question 4
The total of a set of memory scores is:
$$432$$
and:
$$N=18$$
Use:
$$\bar{x}=\frac{\sum x}{N}$$
to calculate the mean. Show your substitution.[2 marks]
Question 5
In an observation study, cooperative behaviour occurred:
$$27$$
times out of:
$$90$$
recorded behaviours.
Use:
$$\text{Percentage}=\frac{\text{Frequency}}{\text{Total}}\times100$$
to calculate the percentage of cooperative behaviours.[2 marks]
Question 6
A researcher reports that:
$$35\%$$
of a sample of:
$$120$$
participants selected Option A.
Calculate the number of participants who selected Option A.[2 marks]
Question 7
A study has a mean score of:
$$14$$
and contains:
$$32$$
scores.
Use:
$$\sum x=\bar{x}\times N$$
to calculate the total of the scores.[2 marks]
Question 8
The total of a set of scores is:
$$684$$
and the mean is:
$$19$$
Use:
$$N=\frac{\sum x}{\bar{x}}$$
to calculate the number of scores.[2 marks]
Question 9
For one cell in a contingency table:
$$\text{Row total}=45$$
$$\text{Column total}=32$$
$$\text{Overall total}=80$$
Use:
$$E=\frac{\text{Row total}\times\text{Column total}}{\text{Overall total}}$$
to calculate the expected frequency.[3 marks]
Question 10
For one cell in a Chi-squared calculation:
$$O=21$$
and:
$$E=16$$
Calculate:
$$\frac{(O-E)^2}{E}$$
Give your answer to three significant figures.[3 marks]
Question 11
A contingency table contains:
three data rows;
five data columns.
Use:
$$df=(r-1)(c-1)$$
to calculate the degrees of freedom.[2 marks]
Question 12
A Chi-squared table has:
$$df=6$$
and:
$$r=3$$
Use:
$$df=(r-1)(c-1)$$
to calculate the number of data columns. Show your working.[3 marks]
Question 13
A researcher calculates a mean response time of:
$$1850\text{ milliseconds}$$
Convert this value into seconds.[2 marks]
Question 14
A correlation coefficient is calculated as:
$$r_s=-0.73642$$
Report the coefficient to three significant figures.[1 mark]
Question 15
A student divides:
$$49\,200$$
by:
$$2050$$
and obtains:
$$240$$
Use an order-of-magnitude calculation to explain why this answer is unreasonable.[3 marks]
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